Jacobi Theta Function
Definition
The function defined as follows $\vartheta$ is called the Jacobi theta Function. $$ \vartheta (\tau) := \sum_{n \in \mathbb{Z}} e^{-\pi n^{2} \tau } $$
Description
While Jacobi functions can originally be more generally defined, it is common to use a special form of them depending on the needs. Note that the Jacobi theta function introduced here does not cover all contexts in its exact meaning.
The following property is especially used in deriving equations that are central to the study of the Riemann zeta function.
Theorem
$$ \vartheta ( \tau ) = \sqrt{ {{ 1 } \over { \tau }}} \vartheta \left( {{ 1 } \over { \tau }} \right) $$
Proof
By defining $f$ as $$ f(n) := e^{-\pi n^{2} \tau} $$ and then $$ \begin{align*} \vartheta ( \tau ) =& \sum_{n \in \mathbb{Z}} e^{ - \pi n^{2} x } \\ =& \sum_{n \in \mathbb{Z}} f (n) \end{align*} $$
Poisson summation formula: $$ \sum_{n \in \mathbb{Z}} f(n) = \sum_{k \in \mathbb{Z}} \widehat{f}(k) $$
According to the Poisson summation formula $$ \begin{align*} \vartheta ( \tau ) =& \sum_{n \in \mathbb{Z}} f (n) \\ =& \sum_{k \in \mathbb{Z}} \widehat{f}(k) \\ =& \sum_{k \in \mathbb{Z}} \mathcal{F} \left[ e^{- \pi \tau n^{2}} \right] (k) \end{align*} $$
Fourier transform of the Gaussian function: $$ \left( \mathcal{F} f \right) (\gamma) = \mathcal{F} \left[ e^{-A x^{2}} \right] (\gamma) = \sqrt{{ \pi } \over { A }} e^{ - {{ \pi^{2} \gamma^{2} } \over { A }}} $$
Considering $x = n$ and $A = \pi \tau$, the Fourier transform of the Gaussian function $e^{ -\pi \tau n^{2}}$ gives $\gamma = k$ as $$ \begin{align*} \vartheta (\tau) =& \sum_{k \in \mathbb{Z}} \mathcal{F} \left[ e^{- \pi \tau n^{2}} \right] (k) \\ =& \sum_{k \in \mathbb{Z}} \sqrt{{ \pi } \over { \pi \tau }} e^{ - {{ \pi^{2} k^{2} } \over { \pi \tau }}} \\ =& \sqrt{{{ 1 } \over { \tau }}} \sum_{k \in \mathbb{Z}} e^{-\pi k^{2} {{ 1 } \over { \tau }}} \\ =& \sqrt{{{ 1 } \over { \tau }}} \vartheta \left( {{ 1 } \over { \tau }} \right) \end{align*} $$
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